The nearness problems for symmetric matrix with a submatrix constraint

In this paper, we first give the representation of the general solution of the following least-squares problem (LSP): given a matrix X∈Rn×pX∈Rn×p and symmetric matrices B∈Rp×pB∈Rp×p, A0∈Rr×rA0∈Rr×r, find an n×nn×n symmetric matrix A   such that ∥XTAX-B∥=min,s.t.A([1,r])=A0, where A([1,r])A([1,r]) is the r×rr×r leading principal submatrix of the matrix A  . We then consider a best approximation problem: given an n×nn×n symmetric matrix A˜ with A˜([1,r])=A0, find A^∈SE such that ∥A˜-A^∥=minA∈SE∥A˜-A∥, where SESE is the solution set of LSP. We show that the best approximation solution A^ is unique and derive an explicit formula for it.